In the age of Machine Intelligence and computerized decision making, we have to deal with subjective imprecision inherently associated with human perception and described in natural language and uncertainty captured in the form of randomness. This treatise develops the fundamentals and methodology of Axiomatic Fuzzy Sets (AFS), in which fuzzy sets and probability are treated in a unified and coherent fashion. It offers an efficient framework that bridges real world problems with abstract constructs of mathematics and human interpretation capabilities cast in the setting of fuzzy sets.

In the self-contained volume, the reader is exposed to the AFS being treated not only as a rigorous mathematical theory but also as a flexible development methodology for the development of intelligent systems.

The way in which the theory is exposed helps reveal and stress linkages between the fundamentals and well-delineated and sound design practices of practical relevance. The algorithms being presented in a detailed manner are carefully illustrated through numeric examples available in the realm of design and analysis of information systems.

The material can be found equally advantageous to the readers involved in the theory and practice of fuzzy sets as well as those interested in mathematics, rough sets, granular computing, formal concept analysis, and the use of probabilistic methods.

Klappentext

It is well known that "fuzziness"-informationgranulesand fuzzy sets as one of its formal manifestations- is one of important characteristics of human cognitionandcomprehensionofreality. Fuzzy phenomena existinnature and are encountered quite vividly within human society. The notion of a fuzzy set has been introduced by L. A. , Zadeh in 1965 in order to formalize human concepts, in connection with the representation of human natural language and computing with words. Fuzzy sets and fuzzy logic are used for mod- ing imprecise modes of reasoning that play a pivotal role in the remarkable human abilities to make rational decisions in an environment a?ected by - certainty and imprecision. A growing number of applications of fuzzy sets originated from the "empirical-semantic" approach. From this perspective, we were focused on some practical interpretations of fuzzy sets rather than being oriented towards investigations of the underlying mathematical str- tures of fuzzy sets themselves. For instance, in the context of control theory where fuzzy sets have played an interesting and practically relevant function, the practical facet of fuzzy sets has been stressed quite signi?cantly. However, fuzzy sets can be sought as an abstract concept with all formal underpinnings stemming from this more formal perspective. In the context of applications, it is worth underlying that membership functions do not convey the same meaning at the operational level when being cast in various contexts.

Inhalt

Required Preliminary Mathematical Knowledge.- Fundamentals.- Lattices.- Methodology and Mathematical Framework of AFS Theory.- Boolean Matrices and Binary Relations.- AFS Logic, AFS Structure and Coherence Membership Functions.- AFS Algebras and Their Representations of Membership Degrees.- Applications of AFS Theory.- AFS Fuzzy Rough Sets.- AFS Topology and Its Applications.- AFS Formal Concept and AFS Fuzzy Formal Concept Analysis.- AFS Fuzzy Clustering Analysis.- AFS Fuzzy Classifiers.